Gradient on product manifold

Question

Let $(M_1,g_1) \times (M_2,g_2)$ be a Riemannian product-manifold, and let $f:(M_1 \times M_2) \rightarrow \mathbb{R}^+$ a positive scalar function on the product manifold, with $f=f_1+f_2$ where each is a function on its individual manifold. can I write $grad(f)=grad(f_1)+grad(f_2)$?

Answer 1

If you want to be precise, some $\pi_i$'s should appear. Here's a full computation: for $f_i\colon M_i \to \Bbb R$, $i=1,2$, and $f\colon M_1\times M_2 \to \Bbb R$ given by $f(x,y) = f_1(x) + f_2(y)$, we have that $${\rm d}f_{(x,y)}(v,w) = {\rm d}(f_1)_x(v) + {\rm d}(f_2)_y(w),$$so if $g$ denotes the product metric, this becomes $$g_{(x,y)}(\nabla f(x,y), (v,w)) = (g_1)_x(\nabla f_1(x), v) + (g_2)_y(\nabla f_2(y), w) = g_{(x,y)}((\nabla f_1(x), \nabla f_2(y)), (v,w)).$$Since $(v,w)$ is arbitrary, we have that $\nabla f(x,y) = (\nabla f_1(x),\nabla f_2(y))$. Without points, this reads $$\nabla f = ((\nabla f_1)\circ \pi_1, (\nabla f_2)\circ \pi_2),$$where $\pi_i\colon M_1\times M_2 \to M_i$, $i=1,2$, are the projections.